Proc. Nati Acad. Sci. USAVol. 80, pp. 2412-2416, April 1983Physics

Intrinsic randomness and intrinsic irreversibility in classicaldynamical systems

(second law/entropy or MC theorem/Markov process)

M. COURBAGE AND I. PRIGOGINEtFacult6 des Sciences, Campus Plaine UniversitM Libre de Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium

Contributed by I. Prigogine, November 22, 1982

ABSTRACT We continue our previous work on dynamic "in-trinsically random" systems for which we can derive dissipativeMarkov processes through a one-to-one change of representation.For these systems, the unitary group of evolution can be trans-

formed in this way into two distinct Markov processes leading to

equilibrium for either to- + Xo or to- - ox. To lift the degeneracy,we first formulate the second principle as a selection rule that ismeaningful in intrinsically random systems. For these systems,

this excludes a set of unrealizable states. As a result of this ex-

clusion, permitted initial conditions correspond to a set of states

that is not invariant through velocity inversion. In this way, thetime-reversal symmetry of dynamics is broken and these systemsacquire a new feature we may call "intrinsic irreversibility." Theset of admitted initial conditions can be characterized by an en-

tropy displaying the amount of information necessary for theirpreparation. The initial conditions selected by the second law cor-

respond to a finite amount of information, while the initial con-

ditions that are rejected correspond to an infinite amount of in-formation and are therefore "impossible." We believe that our

formulation permits a microscopic formulation of the second lawof thermodynamics for well-defined classes of dynamical systems.

1. Introduction

The second law of thermodynamics has on the macroscopic leveltwo distinct aspects: The first is a "negative" aspect limiting ourmanipulations of matter (impossibility of a perpetuum mobileof the second kind, etc.). As a result of this limitation, thereappears an arrow of time on the macroscopic level. The secondaspect of the second law is a "positive" or "constructive" onepostulating the existence of entropy that tends to a maximumat equilibrium.

Is it possible to incorporate these two aspects into the frameof classical dynamics and to formulate the second law of ther-modynamics on the microscopic level?We (with B. Misra) have shown (1) that, for highly unstable

classical dynamical systems, there exists a nonunitary equiva-lence between the unitary group of operators governing theevolution of the distribution functions and Markov processesdescribed by the strictly contracting semigroup. For such dy-namical systems, the deterministic dynamics can therefore betransformed into a stochastic markovian process simply througha "change of representation" that involves no loss of informa-tion. We called such systems intrinsically random. The impor-taace of nonunitary transformations is stressed in our earlierwork in connection with the foundations of nonequilibrium sta-tistical mechanics (2).

However, because of the time reversibility of the dynamicmotion, it seems at first that there would always exist two changes

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of representation, one leading to a strictly contractive semi-

group corresponding to approach to equilibrium for t -+ +00

and the other for t -X. How can this degeneracy be lifted?

In a recent paper, Misra and Prigogine (3) presented a fewqualitative remarks concerning this problem, writing "Thephysical origin of the symmetry breaking in question is a lim-itation on physically observable states. Such a limitation comes,in the first place, from (strong) instability of dynamical motionas a consequence of which the concept of phase space trajec-tories ceases to be physically meaningful and the physically re-

alizable states of the system need to be described in terms of(Gibbs) distribution functions. But the existence of symmetrybreaking under consideration is the expression of a further lim-itation: not all distributions but only a suitable proper subset ofthem can correspond to physically realizable states."We now want to elaborate these remarks and to show that the

limitation on physically observable states leads indeed to break-ing of the time symmetry of intrinsically random dynamical sys-

tems.To illustrate our point of view it may be useful to refer first to

the well-known example of symmetry-breaking bifurcations. Itis well known that solutions of differential equations may be lesssymmetric than the equations themselves (for example, we may

have solutions that are "left" and others that are "right" while thedifferential equation is invariant by the space reflection r -> -r).

It should also be noted that the existence of bifurcations is ob-viously not a universal property of differential equations. Simi-larly, we have to look for specific properties of dynamical sys-tems to give a dynamic formulation to irreversibility.

One usually considers two realizations of classical dynamicalgroups. The first is as point transformations St on the phase spacer or as trajectories; this is the classical Newton-Hamiltonianpoint of view. The second corresponds to a unitary group ofoperators Ut acting on square integrable distribution functionsL2(F) according to

(Utp) (W) p (S-tw); [1.1]

this can be called the Gibbs-Koopmann point of view. Both re-

alizations are "universal;" they can be applied to arbitrary dy-namical systems. Both also preserve the maximum symmetry ofthe dynamical groups. However, new highly nontrivial featuresappear in strongly unstable dynamical systems such as K flows.First, as mentioned above, they are "intrinsically random."

This means that the unitary group U, is transformed througha bounded nonunitary linear operator A on L into a strict semi-group of contractive operators of Markov process W:t that sat-

isfies the following relation (1):

W*= AUtA- [L.2]

t Also at Center for Studies in Statistical Mechanics, University of Tex-as, Austin, TX.

2412

Proc. Natl. Acad. Sci. USA 80 (1983) 2413

with a We theorem for the functional frot log Pt du, where Pt- Wp and P = A p.Two distinct transformations A+ and A and two distinct

semigroups W* and Wt* for t . 0 and t ' 0, respectively, sat-isfy the above relation. This is the degeneracy to which we al-luded briefly above.

But there is also a second element that appears with con-sideration of strongly unstable systems. Let us introduce a groupof automorphisms at on the set of all probability measures onthe phase-space that transform a measure v into atv = vt:

vt(E) = (atv) (E) = v(S-tE) [1.3]for any measurable set E. Now, for highly unstable systems, wecan consider states and measures that have broken time sym-metry. Examples are measures concentrated on dilating fibersand contracting fibers as they appear in all K-systems or hy-perbolic systems (see also ref. 4). Let us illustrate the meaningof fibers in the so-called baker's transformation that sends a point(p,q) of the unit square F into B(p,q):

B(pq) = 2p, ) 0 'p < 1/2

B(p,q)= (2p-1,q+1/2) 1/2 P<. [1.4]

To see the action of B on any subset of r, one has to squeezethe square in the direction of q, stretch in the direction of p,

and finally stack the right half on the left half.Contracting and dilating fibers are horizontal and vertical lines

of the square. The measures concentrated on contracting fiberswill have no limit for t -> +00, as is also the situation in the caseof Dirac measures concentrated on points, and the measuresconcentrated on dilating fibers will tend to the uniform mea-sure-i.e., the Lebesgue measure-for t -> +o. But for t --

-00, dilating fibers again have no limit. Therefore, we have twoclasses of states that are not invariant with respect to the timeinversion t -+ t.Of course the baker's transformation is too symmetrical to

allow one to choose one class of states and exclude the other.But dilating and contracting fibers acquire an important phys-ical significance in other models, such as the Lorentz gas.

This system represents the motion of an infinite number ofnoninteracting particles. Each moves with constant velocity andreflects elastically from periodically distributed scatterers. Thesystem is reduced to the motion of a billiard ball given by thetransformation that associates to each colliding arrow the nextone. Sinai (5) has shown that this is a K-system, and Gallavottiand Ornstein (6) have shown that it is furthermore Bernoulli.The fibers are given as a limit of k-order fibers: a smooth pieceof the k-order contracting fiber yk(x) containing the arrow x isthe set of all arrows that will be parallel after executing k col-lisions and k-order dilating fiber yd(x) is the set of arrows thatwas parallel before k collisions. The dilating and contracting fi-bers are the limit of y and yk when k -- 0.

In summary, intrinsically random systems are characterizedby realizations of dynamics that have a lower symmetry thanthe maximum symmetry of dynamical groups. It is for this rea-

son that they may play a fundamental role in the formulationof irreversibility on the microscopic level.

2. Microscopic formulation of the second law ofthermodynamics

We can now express the second law of thermodynamics on themicroscopic level by requiring that only measures that lead to

equilibrium for t -- +oo are observed in nature or can be pre-

pared in the laboratory.

Therefore, we accept probability measures concentrated ondilating fibers and leading to equilibrium for t-4 +oo but rejectmeasures concentrated on single or countable contracting fi-bers. We also exclude measures concentrated on points inagreement with the idea that trajectories become impossibleidealizations in the case of highly unstable systems.

In situations such as the Lorentz gas, this selection rule ex-presses the unrealizability of experiences in which a set of par-ticles that undergo several collisions will asymptotically emergewith parallel velocities. In contrast, it is possible to prepare an"ensemble" of particles hitting an obstacle with parallel veloc-ities-i.e., a beam of particles. The selection of this class ofinitial conditions is compatible with dynamics as it is propa-gated in time. That is, the measures concentrated on finite orcountable numbers of dilating fibers are mapped by the auto-morphism into measures of the same class (see below).

This formulation of the second law lifts the degeneracy in thetransition to Markov processes through the nonunitary trans-formation.

In fact, we shall show that a A transformation can be associ-ated not only with square integrable distribution functions butalso for singular probability measures such as measures on fi-bers. The two transformations A+ and A are no longer acting onthe same time-symmetric set of states but on two distinct sets ofstates. The selection rule between these states leads to a uniquechoice between the two Markov processes A+atA-' (t . 0) andA-atAi1 (t ' 0). Intrinsically random systems for which such aphysical distinction between contracting and dilating fibers canbe made will be called intrinsically irreversible.

As the contracting and the dilating fibers are velocity in-verses of each other, our microscopic formulation of the secondlaw (i.e., as a selection rule) expresses that the set of probabilitymeasures (or nonequilibrium ensembles) that can be realizedthrough suitable experiments is not invariant in respect to ve-locity inversion.

It should be noted that in our formulation the second law isnot a consequence of dynamics. It involves a supplementaryphenomenological element (the impossibility of realizing a setof asymptotically converging trajectories). Again the situationis similar to that in macroscopic physics as it involves a limi-tation. Moreover, there appears, as expected, a close link: onlythe unstable dynamical systems we called intrinsically randomcan in addition be intrinsically irreversible. The analogy withthe case of differential equations in which bifurcation is a gen-eral but not universal phenomenon is striking.

Finally, the selection rule is also related to the entropy as-sociated with states. As we will show below, the entropy as-sociated with dilating fibers is finite while that associated withcontracting fibers is infinite. To realize a distribution on con-tracting fibers, one needs an infinite amount of information.This is due to the fact that any small error in the knowledge ofthe contracting fibers will be exponentially amplified and in-finite information is necessary to describe this type of initialcondition. The exclusion of contracting fibers has therefore aphysical meaning.

It will be shown elsewhere that similar considerations alsoapply to large systems involving an infinite number of degreesof freedom.

3. Singular measures and the arrow of time

Let us consider an abstract classical dynamical system corre-sponding to phase space r, a normalized measure AL on a Borelo-algebra W of measurable subsets of r and a group of measurepreserving transformations St. In addition to the normalizedmeasure A on A, we consider nonequilibrium measures v thatevolve under the automorphisms group at acting on the set of

Physics: Courbage and Prigogine

2414 Physics: Courbage and Prigogine

all normalized positive measures of r (see Eq. 1.3).For what will follow, it is necessary to introduce the time

inversion transformation I, which is a continuous one-to-one

transformation onto F such that (i) la = 1, (ii) ISJ = St1 = S,and (iii) (IE) = j.(E) for all E. This transformation defines theoperator V on the- space of measures

(Vv) (E) = V' (E) = v(IE). [3.1]

The transformation I corresponds in classical systems to veloc-ity inversion.We now define the approach to equilibrium: vt approaches

equilibrium (i.e., vt-t opt) if for all continuous functions:

f dvt v+)f d;. [3.2]t r. 0

denoted 40, is called the K partition. First of all, we define thefunctional [1 (v, D-) for an arbitrary measure v and a finite par-

tition a:

a(v,a) = E lo(Q)c4 V(Q)] [4.3]

where ¢(x) = x log x fk(v, 9.) can be interpreted as the relativeinformation of v with respect to 1L, both restricted to the finiteexperiment (Qj, , Qn).The convexity of the function ¢)(x) implies the positivity of

fk(v, 91). We can also verify that fl(v, 9P,) is nondecreasing as

9P, becomes finer so that fl(v, 91nJ tends, for n -- o0, either to

a finite limit or to +Xo. The entropy of v relative to the o--al-gebra tO is the limit

Cl(V, to) = lim fl(v, iP,)n--ooo [4.4]

It may occur that some measures approach equilibrium forboth t -+ ±00, or only for t -- +00 or t --, or for neither

t-o+ ±00.

As a consequence of the definition of mixing, any initial nor-malized measure that is absolutely continuous with respect to,u approaches equilibrium both in the past and in the future.

Therefore, to distinguish between past and future, one needsto consider probability measures having a nonvanishing sin-gular component with respect to 1L.

Let us recall. that a measure v is concentrated over a set AEE W if the complement of A, A, is of null v measure. A is thena support of v. A-measure is singular with respect to v if Atand v have disjoint supports-that is, if is concentrated on

a set having null v measure.We now consider dynamical systems for which there exist

two classes of probability measures X and (-, where E (Eiff ftt -- a for t -+ 00 and not for t -m -00 and A E L' iff ft

-*u for to- -00 and not for to- +00. It is clear that the velocityinversion is a one-to-one mapping between A and (E-: V =

IM-. It is also clear that both ft and its velocity inverse At' can-

not belong to the same class. Finally the class A is propagatedby dynamics-i.e., atV+ C '

as it follows from the definition.The use of singular measures is of interest in connection withvarious problems, as will be shown elsewhere. Here the mainpoint is that our formulation of the second law as a selectionprinciple excludes all measures of the class (-.

4. Entropy as a selector of initial conditions

Let us consider a K system (F, 1, tL, St) that has a distinct sub-cr-algebra 21o C W that satisfies the following properties:

(i) Stf1o = 2t C %t' if t s t' [4.1]

(ii) U Wt = [42]

and

(iii) nf 2t = %-x

is the trivial sub-ar-algebra generated by F and all sets of nullmeasures.

For simplicity, we consider here a discrete time parameter,but all results can be extended to continuous time flows withoutchange (we denote St = Bt).

As (F, %O, IL) has the structure of a Lebesgue space iso-morphic to the Lebesgue measure on (0, 1), there exists a de-creasing sequence of finite partitions PP,, such that A1+ is a re-

finement of OPn (we write 9n+1 2 OP,,) and lim,, 9Pn is a partitionof F formed by an uncountable number of cells, with each hav-ing measure zero and generating %O. This partition, generally

This entropy takes an infinite value for measures concen-

trated on contracting fibers that are the elements of the K par-

tition do and it takes a finite value for all measures concentratedon dilating fibers that are the elements of 40 = I 4), where I isthe velocity inversion. One can check by a direct calculationthese results for the baker's transformation (see Appendix). Theyare, however, quite general. Indeed, they result from the fol-lowing proposition:PROPOSMION. If liMos C(v, 9in) <+00, then, the measure

V%,o restriction of v to the sub-ar-algebra 21o, is absolutely con-

tinuous with respect to 1eO§ and its density po satisfies to frPologpdik = fl(v, Wo). Conversely, if veto is absolutely continuouswith respect to pW.0 with density po and fr polog podtk < +00,then

fl(v, Wo) = fpologpodA. [4.5]

The important point of this proposition is that a measure vcan be singular with respect to on W while its restriction toa sub-vr-algebra tO can be absolutely continuous with respectto the restriction pet.Now let vbe a uniform measure concentrated on one dilating

fiber-that is to say, adz = UlIwo. Then according to the secondpart of the proposition, fk(v, TO) = 0 for po = 1. Conversely,by using the first part of the proposition, we see that, when themeasure v is concentrated on a contracting fiber, vist is singularwith respect to kW0 and limnd flQ(v, 9Pn) = +00.

As a result of me above proposition, one sees that all mea-sures of Dirac type (phase points) have infinite entropy.

In the case of distributions p satisfying f p2dp <+0,fl(VoW1) is identical to the entropy, which has been introduced inthe work of Misra and Prigogine (7).

n(Vp, W1) = f Pop log Pop dI, [4.6]

where Po is the projection onto the subspace L? (Wo) 5 Le Thisresults from the above proposition.The entropy fl(v, W1) is now the extension to all probability

distributions of the entropy (Eq. 4.6) previously introduced forp E L2. It is now a selector of the initial conditions as it satisfiesthe following requirements: (i) [1(v, to) = +00 for all distri-butions v concentrated on the contracting fibers and phase points,(ii) ft(v, Wo) < +0 for distributions concentrated on the dilat-ing fibers, (iii) i(Vt, 2W1) is decreasing to zero for any vO suchthat 1i(it, Wo) < +0, and (iv) if fk(vo, Wo) < +00, then Vt tends

§ Recall that xw0 is the measure on WO given by the relation vlwo(E) =v(E) for any E E Wo.

Proc. Natl. Acad. Sci. USA 80 (1983)

Proc. Natl. Acad. Sci. USA 80 (1983) 2415

to equilibrium ji as t -* +oo. i and ii express the fact that theinformation needed to realize a distribution on a dilating fiberis finite while the information needed to realize its velocity in-verse is infinite. As soon as this entropy is finite, it tends mono-tonically to the entropy of the equilibrium in the future and,more important, this implies that the initial distribution willtend to tk as t -) +00 (property iv).

Classical K systems are generally considered as nondeter-ministic ergodic systems mainly on account of the strict posi-tivity of the K-S-entropy. However, in addition, classical dy-namical K systems have a most remarkable property that isreflected by the structure of their state space. This space con-tains one distinct set (the dilating fibers) that is physically re-alizable while its velocity inverse (the contracting fibers) is not.It is clear that the entropy f2(v, 20) is related to the choice of2o. If instead of %O we consider I %O, then we get the oppositeresult. The main point is that both classes of fibers cannot cor-respond to finite entropy and therefore the consideration of fi-bers permits us to distinguish between the two semigroups.

Although 11(vt, %O) is a Liapounov function, it represents onlya component of the entropy, since it remains constant as soonas the distribution v is equal to it on the sub-v-algebra 2O ofA; in other words, M(v, %O) takes the minimum value 0 whenthe expectations frfdv of the subclass of observables f, whichare coarse grained with respect to %O, have the equilibrium val-ues. However, as emphasized in our previous work, the theorywe are developing is not a coarse-graining theory but takes intoaccount the whole information contained in the initial distri-bution v. The entropy is the result of a transformation A of theinitial probability distribution vo into a distribution Po given onthe same ar-algebra of observable events as vo-that is, withoutany contraction of the initial o-algebra W. Moreover, the trans-formation is one-to-one. The effect of this change of represen-tation is to transform the deterministic automorphisms groupat into a Markov process. The entropy is then expressed in termsof this transformed distribution probability evolving under theMarkov process.The point we want to emphasize is that our transformation

has to act on the set of states selected by the second law, whichcontains, in addition to the set of states of type, which is in-variant by the time inversion, a class of states that is not time-inversion invariant and compatible with dynamics. The A trans-formation is essentially the extension of the transformation Apreviously constructed in L . The selection rule on the set ofinitial conditions therefore becomes a selection rule betweenthe two possible semigroups of Markov process.

5. Transition to a Markov process

We consider a classical dynamical system (F, W, St, ,u) realizedas a group of automorphisms-at (see Eq. 1.3) on the set of allprobability measures (or states) on (F, 2). Such a systemis calledintrinsically random and intrinsically irreversible if it meets thefollowing conditions. (A) There exists a class of states + thathas the properties (i) C is physically realizable, (ii) the velocityinverse (S = V ( is not physically realizable, (iii) at(+ CC +,and (iv) a tends to ,u for t -- +oo but at(- does not. (B)There exists a one-to-one affine transformation A on a subsetW of states that containsV but may not contain (-, with thefollowing properties (i') A maps a state into a state; (ii') A is one-to-one; (iii') atS5+ C Sl+ and at~b+ tends to ,u for t-3 +00; (iv')Aatv = WI'Av for any v cE W, where W* is a semigroup ofa Markov process associated with a transition probability P(t,sw,E)-i.e., if v

= A vthen,

(W%*v)(E) = f rdi'(w)P(t,w,E) [5.1]

for any E E W and (v) AvP is absolutely continuous with a den-sity At verifying a be theorem-i.e.,

l(Pt) = rjtlog ptdA [5.2]decreases monotonically and tends to zero as t --* m.

As emphasized in the Introduction, singular measures con-centrated uniformly on finite or countable numbers of dilatingfibers play the role of A+ for K systems. Let Qbl be the set ofall measures v satisfying

[5.3]Using the results of Section 4 shows that a measure of this

class has absolute continuous restriction to Wi with density pi.Let Si be a sequence of positive numbers such that >2j8, = 1.We can show that I8ipi converges to an integrable function p.We can now introduce A as follows:

A:vE2+ Av=vwith(Av) (E)=PdW. [5.4]

It is clear that A maps the normalized states into a normal-ized state and that the set of initial conditions (see Eq. 5.3) onwhich A acts is propagated by the dynamics at. One can showthat A is affine and one-to-one.Now choose the sequence Si such that A, = E8,, satisfies

the requirement

Aj+1is a decreasing sequence.

We recall that, in our previous work (1, 8, 9), we consideredthe semigroup:

W* = AUtA-',where A is the bounded operator on L,;

A = I SiPi

Pip = E"(p), p E L2.

[5.6][5.7]

We have shown that the operator W, defines a transitionprobability of a Markov process with

P(t, w, E) = (Wt'pE) (w), [5.8]where pE is the characteristic function of the subset E.

Ifwe consider a measure v of 21), we can show the followingrelation:

(A atv) (E) = f (Wt pE)(w)Ao() dA [5.9]

for all E E 1. Since Av, is absolutely continuous with densityPt, we have

(A atv) (E) = |f du. [5.10]

The family W, is a semigroup of contracting positivity-pre-serving operators on L', and its adjoint on L' is W*. Eqs. 5.9and 5.10 therefore imply that

At = W:' po

26' is then the subset of 26, such that

I Aolog podA <

[5.11]

+00. [5.12]

Physics: Courbage and Prigogine

fl(v,%j) < +00 (i= ...,-1)0,+l).

[ses]

2416 Physics: Courbage and Prigogine P

and the W theorem is satisfied by any v E a'+.Mathematical details of the proof of the above properties will

be published elsewhere. We want to show here, as an example,that the singular distributions uniformly concentrated on onedilating fiber in the case of the baker's transformation are offinite entropy (Eq. 5.12). Because of the following inequality

folog9o f p2 dL - 1, [5.13]

it is enough to show that fr p2di < +0. This will be verifiedin the Appendix.The A transformation has a surprising property: it appears as

a regularization of a class of singular measures that become,through a change of representation, absolutely continuous andsatisfy a We theorem.Of course, another A transformation can be constructed for

t < 0; nevertheless, it acts now on LY. Again the selection rulelifts the degeneracy.

6. Conclusions

We believe that we have presented in this paper a rigorous for-mulation of the second law on the microscopic level.

It may appear unexpected that we need singular measuresto introduce the second law. This comes from the fact that wewant to make a sharp distinction between measures that arepermitted and measures that are not. Of course, even a regularfunction close to a contracting fiber will require such a high in-formation content that it will be practically impossible to re-alize it for a given state of technology.Our interpretation of entropy is quite close to the traditional

one as it expresses, finally, the information content of a non-equilibrium measure in respect to the equilibrium measure. Inthis sense, it is a natural extension of conventional informationtheory based on the use of Markov chains. The new elementis the transition from dynamics to Markov chains and thereforethe dynamical interpretation of information theory.

It should, however, be noted that the popular interpretationof entropy as expressing an improbability and not an impos-sibility (see, for example, ref. 10) can be viewed in a new light.It is as the result of the exclusion of classes of initial conditionsin highly unstable systems that the probabilistic interpretationof dynamics becomes possible. In a sense, our approach putsthe usual interpretation of entropy upside down.

Also in this perspective, the classical distinction between ini-tial conditions and dynamical laws loses much of its meaning.Indeed the nonequilibrium entropy, which is a function of thedynamics of the system, permits us to associate to each initialcondition a measure corresponding to its "information content."This shows once more how much a description of nature in-cluding the second law of thermodynamics deviates from theclassical dynamical description (see, for example, ref. 11).

Appendix

The o-algebra of Borel sets of the baker's transformation can bedescribed as follows:

Let QP be a partition of F formed by two rectangles Po ={w E r:p < 1/2} and P1 = {w:p > 1/2}. B"QP will denote the parti-tion {B Po, BnP1}. The intersections of the form A":::::i+k.1 -

BnPi1 n ... n B1n+k- Pi, are called cylinders. These are the cellsof the partition of F denotedV1rkA The partitionV° n+l BigP tends to to = V%X B!P formed by vertical lines. Thispartition generates a a-algebra denoted, Wo. Similarly, V'j Bngtends to the partition {O formed by horizontal line and gen-erates W'. The velocity inversion here is the permutation of p

and q so that 12P = B9P and

Iwo.= wto [A.2]ao is the set of contracting fibers. Similarly, g is formed by

dilating fibers. Each element of f0 is mapped by the transfor-mation B into two horizontal lines.

Let v be a singular measure uniformly concentrated on thedilating fiber q = qo. The restriction of this singular measureto sub-or-algebra Wi is absolutely continuous with respect. to theLebesgue measure. Indeed, not only is the function fl(v, WI)finite, but also the stronger one, namely,

fk2(v, 2t) = lim f12(v, V B'QP), [A.3]

where I2(v, 9l) is given by substitution of Ix12 for ¢(x) in Eq.4.3, is also finite. In that case, f12(v, A,) is related to the densitypi of vylt by f2(v, ,i) = fr pj dA. By straightforward calcu-lations, we check that

nfI2(v, V BP) = 2n (n > 0)

-m= 1 (nO0). [A.4]

To verify that - 8ipi is L2 convergent, we use the fol-lowing relation

+N +N

ESA8p, = A-NP-N + > Ai(pI - Pi-1) - AN+1 PN. [A.5]-N -N

Now, AN+1ifp2dp = A2+1/2" -> Oas n-* o andP-N- 1as N -* 00 Therefore, the convergence is ensured by the con-vergence of the sum

[A.6]

By using the relation E-'-(pi) = pi-1, one has the following re-lation:

E AfI -|P-Pi2dpu = >(Ai -A?+i)t_2(v, Wi)idt ,( i 2 [A.7]

The property of monotonic decrease of An+1/Ak and thed'Alembert test ensure the convergence. Therefore, | polog podo < +Xo. On the other hand, one easily sees that f12(P',Vn B'P) = 2m+1 and therefore f2(v', WlO) = +oo, as we stressedin the proposition.We want to thank Professor B. Misra for inspiring discussions. Part

of this work was supported by the Robert Welch Foundation (Texas).1. Misra, B., Prigogine, I. & Courbage, M. (1979) Physica 98A, 1-

26.2. Prigogine, I., George, C., Henin, F. & Rosenfeld, L. (1973) Chem.

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on Long Time Predictions in Dynamical Systems, eds. Szebehely,V. G., Horton, C. H. & Reichl, L. E. (Wiley, New York).

4. Arnold, V. I. & Avez, A. (1968) Ergodic Problems of ClassicalMechanics (Benjamin, New York).

5. Sinai, Y. G. (1970) Russ. Math. Surv. 25, 137-189.6. Gallavotti, G. & Ornstein, D. S. (1974) Commun. Math. Phys. 38,

83-101.7. Misra, B. & Prigogine, I. (1980) Progr. Theor. Phys. Suppl. 69,

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Proc. Natl. Acad. Sci. USA 80 (1983)

1: A. ipi pi-II' dA.i